Could you be thinking of the Dalai Lama?

You're thinking of a practice which is relatively unique to Vajyarana Buddhism, based historically in Tibet, not Buddhism in general. This sect of Buddhism, which the Dalai Lama and other lamas belong to is very visible in western pop depictions of Buddhism but is by no means universal, or even the largest sect.

As for how the successor is determined, here's an example of how the new Dalai Lama would be selected.

To start the search, High Lamas may have a vision or dream. If the previous Dalai Lama was cremated, they watch the direction of the smoke to indicate the direction of rebirth. They often meditate at Lhamo La-Tso, central Tibet's holy lake, and wait for a vision or indication of the direction in which to search. This relates to a belief that the female guardian spirit of the lake promised the first Dalai Lama that she would protect the reincarnation lineage. When these visions have been followed up and a boy found, there are a series of tests to ensure that he is the rebirth. There is a secret set of criteria against which the child is assessed. In addition to this, the main test consists of presenting the boy with a number of items to see if he can select those which belonged to the previous Dalai Lama.

As I said none of the above would be relevant in most Buddhist cultures outside Tibet and Central Asia.